Growth of the number of geodesics between points and insecurity for riemannian manifolds

A Riemannian manifold is said to be uniformly secure if there is a finite number $s$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $s$ point obstacles. We prove that the number of geodesics with length $\leq T$ between every pair of points in a uniformly secure manifold grows polynomially as $T \to \infty$. We derive from this that a compact Riemannian manifold with no conjugate points whose geodesic flow has positive topological entropy is totally insecure: the geodesics between any pair of points cannot be blocked by a finite number of point obstacles.Comment: 14 pages, 0 figure

On the ergodicity of partially hyperbolic systems

Pugh and Shub have conjectured that essential accessibility implies ergodicity, for a $C^2$, partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satsified by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for $C^{1+\gamma}$ partially hyperbolic systems.Comment: 46 pages, 4 figure

Thermodynamics for geodesic flows of rank 1 surfaces

We investigate the spectrum of Lyapunov exponents for the geodesic flow of a compact rank 1 surface.Comment: 28 pages, 4 figure

Open problems and questions about geodesics

The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi Riemannian and magnetic structures on manifolds.Comment: This is the final version of the pape

The Weil-Petersson geodesic flow is ergodic

We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (in fact Bernoulli) and has finite, positive metric entropy.Comment: 53 pages. Errors corrected in earlier version and some expository material removed. To appear in Annals of Mat

Average pace and horizontal chords

We are motivated by a problem about running: If a race was completed in an average pace of P minutes per mile, is there necessarily some mile of the race that was run in exactly P minutes? The answer is no. We explain why, and describe the history of this celebrated problem, known as the Universal Chord Theorem. We also clarify and streamline the proof of a more powerful result by Heinz Hopf from 1937.Comment: 16 pages including appendix, 6 figure

Formulating and critically examining the assumptions of global 21-cm signal analyses: How to avoid the false troughs that can appear in single spectrum fits

The assumptions inherent to global 21-cm signal analyses are rarely delineated. In this paper, we formulate a general list of suppositions underlying a given claimed detection of the global 21-cm signal. Then, we specify the form of these assumptions for two different analyses: 1) the one performed by the EDGES team showing an absorption trough in brightness temperature that they modeled separately from the sky foreground and 2) a new, so-called Minimum Assumption Analysis (MAA), that makes the most conservative assumptions possible for the signal. We show fits using the EDGES analysis on various beam-weighted foreground simulations from the EDGES latitude with no signal added. Depending on the beam used, these simulations produce large false troughs due to the invalidity of the foreground model to describe the combination of beam chromaticity and the shape of the Galactic plane in the sky, the residuals of which are captured by the ad hoc flattened Gaussian signal model. On the other hand, the MAA provides robust fits by including many spectra at different time bins and allowing any possible 21-cm spectrum to be modeled exactly. We present uncertainty levels and example signal reconstructions found with the MAA for different numbers of time bins. With enough time bins, one can determine the true 21-cm signal with the MAA to $<10$ times the noise level.Comment: 19 pages, 4 figures, accepted to ApJ. Since previous version, added frequency correlation structure under MAA analysis and laid out extension of MAA to motion-induced dipole of 21-cm signa

A new goodness-of-fit statistic and its application to 21-cm cosmology

The reduced chi-squared statistic is a commonly used goodness-of-fit measure, but it cannot easily detect features near the noise level, even when a large amount of data is available. In this paper, we introduce a new goodness-of-fit measure that we name the reduced psi-squared statistic. It probes the two-point correlations in the residuals of a fit, whereas chi-squared accounts for only the absolute values of each residual point, not considering the relationship between these points. The new statistic maintains sensitivity to individual outliers, but is superior to chi-squared in detecting wide, low level features in the presence of a large number of noisy data points. After presenting this new statistic, we show an instance of its use in the context of analyzing radio spectroscopic data for 21-cm cosmology experiments. We perform fits to simulated data with four components: foreground emission, the global 21-cm signal, an idealized instrument systematic, and noise. This example is particularly timely given the ongoing efforts to confirm the first observational result for this signal, where this work found its original motivation. In addition, we release a Python script dubbed $\texttt{psipy}$ which allows for quick, efficient calculation of the reduced psi-squared statistic on arbitrary data arrays, to be applied in any field of study.Comment: 22 pages, 10 figures, submitted to JCAP, psipy code to calculate new statistic available at https://bitbucket.org/ktausch/psip

Rates of mixing for the Weil-Petersson geodesic flow II: exponential mixing in exceptional moduli spaces

We establish exponential mixing for the geodesic flow $\varphi_t\colon T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for ${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the phase space $T^1S$ and rescaling the flow $\varphi_t$.Comment: 42 page

Rates of mixing for the Weil-Petersson geodesic flow I: no rapid mixing in non-exceptional moduli spaces

We show that the rate of mixing of the Weil-Petersson flow on non-exceptional (higher dimensional) moduli spaces of Riemann surfaces is at most polynomial.Comment: 12 pages. 4 figure